Question: The lifespans of snakes in a particular zoo are normally distributed. The average snake lives $27.8$ years; the standard deviation is $5.3$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a snake living less than $43.7$ years.
$27.8$ $22.5$ $33.1$ $17.2$ $38.4$ $11.9$ $43.7$ $99.7\%$ $0.15\%$ $0.15\%$ We know the lifespans are normally distributed with an average lifespan of $27.8$ years. We know the standard deviation is $5.3$ years, so one standard deviation below the mean is $22.5$ years and one standard deviation above the mean is $33.1$ years. Two standard deviations below the mean is $17.2$ years and two standard deviations above the mean is $38.4$ years. Three standard deviations below the mean is $11.9$ years and three standard deviations above the mean is $43.7$ years. We are interested in the probability of a snake living less than $43.7$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the snakes will have lifespans within 3 standard deviations of the average lifespan. The remaining $0.3\%$ of the snakes will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({0.15\%})$ will live less than $11.9$ years and the other half $({0.15\%})$ will live longer than $43.7$ years. The probability of a particular snake living less than $43.7$ years is ${99.7\%} + {0.15\%}$, or $99.85\%$.